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Counterfactual Donkeys Dont Get High Mike Deigan Yale University - PowerPoint PPT Presentation

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Counterfactual Donkeys Dont Get High Mike Deigan Yale University SuB22, Potsdam New Data 1 / 40 New Data Suppose Allie and Bert think Mary the potter probably didnt make anything yesterday. 1 / 40 New Data Suppose Allie and Bert

  1. Ordering Semantics + Dynamic Binding j is an A -possibility for i (or j ∈ / A / i ) i ff ∃ k ( g k = g i ∧ j ∈ { k } [ A ]). Selection function: (8) f ( A , i ) = { j : j ∈ / A / i ∧ ¬∃ k ( k ∈ / A / i ∧ w k < w i w j ) } . Finds the nearest A -possibility, where possibilities are ordered by their worlds. 11 / 40

  2. Ordering Semantics + Dynamic Binding j is an A -possibility for i (or j ∈ / A / i ) i ff ∃ k ( g k = g i ∧ j ∈ { k } [ A ]). Selection function: (8) f ( A , i ) = { j : j ∈ / A / i ∧ ¬∃ k ( k ∈ / A / i ∧ w k < w i w j ) } . Finds the nearest A -possibility, where possibilities are ordered by their worlds. (9) s [ A � C ] = { i : i ∈ s ∧ ∀ j ( j ∈ f ( A , i ) ⊃ { j } [ C ] � ∅ ) } C is verified by all selected possibilities 11 / 40

  3. Ordering Semantics + Dynamic Binding (3) If Balaam owned a donkey, he would beat it. 12 / 40

  4. Ordering Semantics + Dynamic Binding (3) If Balaam owned a donkey, he would beat it. (10) ∃ xDBO ( x ) � BB ( x ) 12 / 40

  5. Dynamic Binding + Ordering Semantics w 0

  6. Dynamic Binding + Ordering Semantics w 0 ¬ DBO DBO DBO = donkey that Balaam owns

  7. Dynamic Binding + Ordering Semantics w 0 ¬ BB BB ¬ DBO DBO DBO = donkey that Balaam owns BB = thing that Balaam beats

  8. Dynamic Binding + Ordering Semantics w 0 • e p • ¬ BB • h BB ¬ DBO DBO DBO = donkey that Balaam owns BB = thing that Balaam beats e = Eeyore h = Herbert p = Platero 13 / 40

  9. w 0 • e • p ¬ BB • h BB ¬ DBO DBO

  10. w 1 ¬ BB BB ∨ w 0 • e • p ¬ BB • h BB ¬ DBO DBO

  11. w 1 • e • p ¬ BB • h BB ∨ w 0 • e • p ¬ BB • h BB ¬ DBO DBO

  12. w 2 • e • p ¬ BB • h BB ∨ w 1 • e • p ¬ BB • h BB ∨ w 0 • e • p ¬ BB • h BB ¬ DBO DBO

  13. w 3 • p • e ¬ BB • h BB ∨ w 2 • e • p ¬ BB • h BB ∨ w 1 • e • p ¬ BB • h BB ∨ w 0 • e • p ¬ BB • h BB ¬ DBO DBO

  14. w 3 • p • e ¬ BB • h BB ∨ w 2 • e • p ¬ BB • h BB ∨ w 1 • e • p ¬ BB • h BB ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � BB ¬ DBO DBO

  15. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � BB ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � BB ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � • h BB ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � BB ¬ DBO DBO

  16. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  17. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  18. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  19. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  20. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  21. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO 14 / 40

  22. w 3 � w 3 , g x → p � w 1 • e • p ¬ BB � w 1 , g x → h � • h BB ¬ DBO DBO ¬ BB DBO

  23. w 3 � w 3 , g x → p � ∃ xDBO ( x ) � BB ( x ) w 1 • e • p ¬ BB � w 1 , g x → h � • h BB ¬ DBO DBO ¬ BB DBO

  24. w 3 � w 3 , g x → p � ∃ xDBO ( x ) � BB ( x ) w 1 • e • p ¬ BB � w 1 , g x → h � ∀ j ( j ∈ f ( A , i ) { j } [ BB ( x )] � ∅ • h BB ¬ DBO DBO ¬ BB DBO

  25. w 3 � w 3 , g x → p � ∃ xDBO ( x ) � BB ( x ) w 1 • e • p ¬ BB � w 1 , g x → h � ∀ j ( j ∈ f ( A , i ) { j } [ BB ( x )] � ∅ • h BB ¬ DBO DBO {� w 1 , g x → h �} [ BB ( x )] ¬ BB DBO

  26. w 3 � w 3 , g x → p � ∃ xDBO ( x ) � BB ( x ) w 1 • e • p ¬ BB � w 1 , g x → h � ∀ j ( j ∈ f ( A , i ) { j } [ BB ( x )] � ∅ • h BB ¬ DBO DBO {� w 1 , g x → h �} [ BB ( x )] ¬ BB w 1 ∈ � BB ( h ) � ? DBO

  27. w 3 � w 3 , g x → p � ∃ xDBO ( x ) � BB ( x ) w 1 • e • p ¬ BB � w 1 , g x → h � ∀ j ( j ∈ f ( A , i ) { j } [ BB ( x )] � ∅ • h BB ¬ DBO DBO {� w 1 , g x → h �} [ BB ( x )] ¬ BB w 1 ∈ � BB ( h ) � ? ✓ DBO

  28. w 3 � w 3 , g x → p � ∃ xDBO ( x ) � BB ( x ) w 1 • e • p ¬ BB � w 1 , g x → h � ∀ j ( j ∈ f ( A , i ) { j } [ BB ( x )] � ∅ • h BB ¬ DBO DBO {� w 1 , g x → h �} [ BB ( x )] = {� w 1 , g x → h �} � ∅ ¬ BB w 1 ∈ � BB ( h ) � ? ✓ DBO

  29. w 3 � w 3 , g x → p � ∃ xDBO ( x ) � BB ( x ) ✓ w 1 • e • p ¬ BB � w 1 , g x → h � ∀ j ( j ∈ f ( A , i ) { j } [ BB ( x )] � ∅ • h BB ¬ DBO DBO {� w 1 , g x → h �} [ BB ( x )] = {� w 1 , g x → h �} � ∅ ¬ BB w 1 ∈ � BB ( h ) � ? ✓ DBO 15 / 40

  30. Ordering Semantics + Dynamic Binding Problem: no universal entailments 16 / 40

  31. Ordering Semantics + Dynamic Binding Problem: no universal entailments (11) If Balaam owned a donkey, he would beat it. (12) a. If Herbert were a donkey Balaam owned, Balaam would beat Herbert. b. If Eeyore were a donkey Balaam owned, Balaam would beat Eeyore. c. If Platero were a donkey Balaam owned, Balaam would beat Platero. 16 / 40

  32. Ordering Semantics + Dynamic Binding Problem: no universal entailments (11) If Balaam owned a donkey, he would beat it. (12) a. If Herbert were a donkey Balaam owned, Balaam would beat Herbert. b. If Eeyore were a donkey Balaam owned, Balaam would beat Eeyore. c. If Platero were a donkey Balaam owned, Balaam would beat Platero. 16 / 40

  33. Ordering Semantics + Dynamic Binding (12-b) If Eeyore were a donkey Balaam owned, Balaam would beat Eeyore. (13) DBO ( e ) � BB ( e ) 17 / 40

  34. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  35. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  36. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB ∨ ∨ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB ∨ ∨ w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO 18 / 40

  37. w 3 w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � BB ¬ DBO DBO � w 1 , g x → p � ¬ BB DBO

  38. w 3 DBO ( e ) � BB ( e ) w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � BB ¬ DBO DBO � w 1 , g x → p � ¬ BB DBO

  39. w 3 DBO ( e ) � BB ( e ) w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � BB ¬ DBO DBO ∀ j ( j ∈ f ( A , i ) { j } [ BB ( e )] � ∅ � w 1 , g x → p � ¬ BB DBO

  40. w 3 DBO ( e ) � BB ( e ) w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � BB ¬ DBO DBO ∀ j ( j ∈ f ( A , i ) { j } [ BB ( e )] � ∅ � w 1 , g x → p � w 2 ∈ � BB ( e ) � ? ¬ BB DBO

  41. w 3 DBO ( e ) � BB ( e ) w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � BB ¬ DBO DBO ∀ j ( j ∈ f ( A , i ) { j } [ BB ( e )] � ∅ � w 1 , g x → p � w 2 ∈ � BB ( e ) � ? ✗ ¬ BB DBO

  42. w 3 DBO ( e ) � BB ( e ) ✗ w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � BB ¬ DBO DBO ∀ j ( j ∈ f ( A , i ) { j } [ BB ( e )] � ∅ � w 1 , g x → p � w 2 ∈ � BB ( e ) � ? ✗ ¬ BB DBO 19 / 40

  43. Ordering Semantics + Dynamic Binding Problem: no universal entailments (14) If Balaam owned a donkey, he would beat it. (15) a. If Herbert were a donkey Balaam owned, Balaam would beat Herbert. b. If Eeyore were a donkey Balaam owned, Balaam would beat Eeyore. c. If Platero were a donkey Balaam owned, Balaam would beat Platero. 20 / 40

  44. Ordering Semantics + Dynamic Binding Problem: no universal entailments (14) If Balaam owned a donkey, he would beat it. (15) a. If Herbert were a donkey Balaam owned, Balaam would beat Herbert. b. If Eeyore were a donkey Balaam owned, Balaam would beat Eeyore. c. If Platero were a donkey Balaam owned, Balaam would beat Platero. How to fix this? 20 / 40

  45. Ordering Semantics + Dynamic Binding Problem: no universal entailments (14) If Balaam owned a donkey, he would beat it. (15) a. If Herbert were a donkey Balaam owned, Balaam would beat Herbert. b. If Eeyore were a donkey Balaam owned, Balaam would beat Eeyore. c. If Platero were a donkey Balaam owned, Balaam would beat Platero. How to fix this? Route 1: special orderings Route 2: high readings 20 / 40

  46. Route 1: Special Orderings Walker and Romero (2015): universal entailments would follow from A � C i ff the closeness ordering were such that 21 / 40

  47. Route 1: Special Orderings Walker and Romero (2015): universal entailments would follow from A � C i ff the closeness ordering were such that for any a , b ∈ D , the closest world which combines with g x → a to form an A -possibility is as close as the closest world which combines with g x → b to form an A -possibility. 21 / 40

  48. Route 1: Special Orderings Walker and Romero (2015): universal entailments would follow from A � C i ff the closeness ordering were such that for any a , b ∈ D , the closest world which combines with g x → a to form an A -possibility is as close as the closest world which combines with g x → b to form an A -possibility. An ordering set S is special relative to a state s and sentence A i ff (16) ∀ i ( i ∈ s ⊃ ∀ j ( j ∈ / A / i ⊃ ∃ k ( k ∈ f ( A , i ) ∧ g j = g k ))) For all possibilities i in s , if j is an A -possibility for i , then among the nearest (relative to i ) A -possibilities is a possibility which shares an assigment with j . 21 / 40

  49. w 3 • p • e ¬ BB • h � w 3 , g x → p � BB ∨ w 2 • e • p ¬ BB • h BB ∨ w 1 • e • p ¬ BB • h BB ∨ w 0 • e • p ¬ BB • h BB ¬ DBO DBO

  50. w 3 • p • e ¬ BB • h � w 3 , g x → p � BB ∨ w 2 • e • p ¬ BB • h BB ∨ w 1 • e • p ¬ BB • h BB ∨ w 0 • e • p ¬ BB • h BB ¬ DBO DBO

  51. w 3 • p • e ¬ BB • h � w 3 , g x → p � BB ∨ w 2 • e • p ¬ BB • h BB ∨ w 1 • e • p ¬ BB • h BB ∨ w 0 • e • p ¬ BB • h BB ¬ DBO DBO

  52. w 3 • p • e ¬ BB • h � w 3 , g x → p � BB ∨ w 2 • e • p ¬ BB • h BB ∨ w 1 • e • p ¬ BB • h BB ∨ w 0 • e • p ¬ BB • h BB ¬ DBO DBO 22 / 40

  53. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB = = w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB = = w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  54. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB = = w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB = = w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO

  55. w 3 • p • e ¬ BB • h � w 3 , g x → e � � w 3 , g x → p � � w 3 , g x → h � = = BB = = w 2 • e • p ¬ BB • h � w 2 , g x → e � � w 2 , g x → p � � w 2 , g x → h � = = BB = = w 1 • e • p ¬ BB � w 1 , g x → e � � w 1 , g x → p � � w 1 , g x → h � = = • h BB ∨ ∨ w 0 • e • p ¬ BB • h � w 0 , g x → e � � w 0 , g x → p � � w 0 , g x → h � = = BB ¬ DBO DBO 23 / 40

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